Large public schools are a far cry from many private schools with regard to electronic record keeping. It's not a priority and doesn't support any direct requirement.
PowerSchool Parent Pertal is used by some small private schools in my area. OP has mentioned that her school chooses not to use it. There are many alternatives now for Student Information Systems, and compared to other classroom costs, it's darn cheap. If this is one of those $40K/year schools, then they're just being lazy.
OP can get together with other parents and petition the Board of Trustees to adopt a system, assuming that such a system is not grossly at odds with the pedagogical philosophy of the school. Of course, if this is one of those schools which touts "it's the learning experience not the grades that matter" (<-- what you get if you're willing to spend over $50K/year), then OP should consider that her educational goals for her child are not well matched with the school her child is enrolled in, and consider transferring.
Effective communication about student progress is important at every level.
That is why the lower level schools in my district use that old technique of having the parents sign and return the graded papers for points. Personally, I thought it wrong that my child could be penalized on her grade for not getting my signature on an assignment where she had an "A". We ALL know that teachers have the parents sign that they have seen the graded assignment/exam so that they don't get an irate parent at the end of the quarter saying, "You never let me know that Johnny/Susey Q. was [failing/not getting an A]." No teacher has ever had an angry parent call and say How dare you not let me know Johnny/Susey Q. was getting A's! I enrolled him/her in summer school to make up the course, and that deposit is non-refundable!"
Even 20 years ago, using old fashioned ExCel, I NEVER had a problem providing the type of grading information that this parent is asking for.
This teacher is either lazy or the Mom is being played by her kid - or both.